L(s) = 1 | − 40.6·2-s − 102.·3-s + 1.13e3·4-s − 769.·5-s + 4.18e3·6-s + 1.56e3·7-s − 2.53e4·8-s − 9.08e3·9-s + 3.12e4·10-s − 1.70e4·11-s − 1.17e5·12-s + 1.41e5·13-s − 6.36e4·14-s + 7.92e4·15-s + 4.48e5·16-s + 3.68e5·18-s − 7.43e5·19-s − 8.75e5·20-s − 1.61e5·21-s + 6.91e5·22-s − 1.55e6·23-s + 2.61e6·24-s − 1.36e6·25-s − 5.76e6·26-s + 2.96e6·27-s + 1.78e6·28-s + 5.73e6·29-s + ⋯ |
L(s) = 1 | − 1.79·2-s − 0.733·3-s + 2.22·4-s − 0.550·5-s + 1.31·6-s + 0.246·7-s − 2.19·8-s − 0.461·9-s + 0.988·10-s − 0.350·11-s − 1.63·12-s + 1.37·13-s − 0.442·14-s + 0.404·15-s + 1.71·16-s + 0.828·18-s − 1.30·19-s − 1.22·20-s − 0.181·21-s + 0.629·22-s − 1.15·23-s + 1.60·24-s − 0.696·25-s − 2.47·26-s + 1.07·27-s + 0.547·28-s + 1.50·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(0.2525577056\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2525577056\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 \) |
good | 2 | \( 1 + 40.6T + 512T^{2} \) |
| 3 | \( 1 + 102.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 769.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.56e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.70e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.41e5T + 1.06e10T^{2} \) |
| 19 | \( 1 + 7.43e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.55e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.73e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.58e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 5.48e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.81e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.79e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.65e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 7.58e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 4.75e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.40e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 7.10e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 7.14e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.40e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.20e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.26e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.75e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.65e8T + 7.60e17T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.51313149184397818935291610887, −9.093492250524772590876587495145, −8.379309535211148962382480222045, −7.72860769235130687373566196311, −6.48489499017694220871561330046, −5.82254246134836961016913087185, −4.11221775810939446637639854847, −2.57149471630148248317117307014, −1.37457233688764081819075192066, −0.32521229414962724500738451347,
0.32521229414962724500738451347, 1.37457233688764081819075192066, 2.57149471630148248317117307014, 4.11221775810939446637639854847, 5.82254246134836961016913087185, 6.48489499017694220871561330046, 7.72860769235130687373566196311, 8.379309535211148962382480222045, 9.093492250524772590876587495145, 10.51313149184397818935291610887